Introduction to Mathematics

• Mathematics is the study of numbers, shapes, quantities, and patterns.
• It is a fundamental science used in various fields like engineering, physics, computer science, and economics.

Core Areas of Mathematics

• Arithmetic: Basic operations (addition, subtraction, multiplication, division) on numbers.
• Algebra: Using symbols to represent numbers and quantities, and solving equations.
• Geometry: Study of shapes, sizes, and positions of figures in space.
• Calculus: Deals with rates of change and accumulation; includes differential and integral calculus.
• Trigonometry: Branch of mathematics dealing with relationships between angles and sides of triangles.
• Statistics: Collecting, analyzing, interpreting, and presenting data.

Numbers

• Natural Numbers: Positive integers (1, 2, 3...).
• Whole Numbers: Natural numbers including zero (0, 1, 2, 3...).
• Integers: Positive and negative whole numbers, including zero (...-2, -1, 0, 1, 2...).
• Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π).
• Real Numbers: Include both rational and irrational numbers.
• Complex Numbers: Numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1).

Algebra Basics

• Variables: Symbols (usually letters) representing unknown values.
• Expressions: Combinations of variables, numbers, and operations.
• Equations: Statements that two expressions are equal.
• Solving Equations: Finding the value(s) of the variable(s) that make the equation true.

Algebraic Operations

• Addition and Subtraction: Combining like terms.
• Multiplication: Distributing terms.
• Division: Simplifying fractions.
• Exponents: Representing repeated multiplication.
• Factoring: Expressing an expression as a product of its factors.

Geometry Fundamentals

• Points, Lines, and Planes: Basic undefined terms in geometry.
• Angles: Formed by two rays sharing a common endpoint (vertex).
• Triangles: Three-sided polygons.
• Quadrilaterals: Four-sided polygons.
• Circles: Set of points equidistant from a center point.

Geometric Formulas

• Area of a rectangle: length × width.
• Area of a triangle: 1/2 × base × height.
• Area of a circle: πr², where r is the radius.
• Circumference of a circle: 2πr.
• Volume of a cube: side³.
• Volume of a sphere: (4/3)πr³.

Trigonometry Principles

• Trigonometric Ratios: Sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), cotangent (cot).
• Right Triangles: Triangles with one 90-degree angle.
• Pythagorean Theorem: a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse.
• Unit Circle: Circle with radius 1 used to define trigonometric functions for all angles.
• Trigonometric Identities: Equations involving trigonometric functions that are true for all values of the variables.

Calculus Concepts

• Limits: Value that a function approaches as the input approaches some value.
• Derivatives: Measure of how a function changes as its input changes.
• Integrals: Represent the area under a curve.
• Fundamental Theorem of Calculus: Connects differentiation and integration.

Differential Calculus

• Differentiation Rules: Power rule, product rule, quotient rule, chain rule.
• Applications of Derivatives: Finding maxima and minima, analyzing rates of change, and optimization problems.

Integral Calculus

• Integration Techniques: Substitution, integration by parts, partial fractions.
• Applications of Integrals: Finding areas, volumes, and average values.

Statistics Basics

• Data Types: Categorical (qualitative) and numerical (quantitative).
• Measures of Central Tendency: Mean, median, and mode.
• Measures of Dispersion: Range, variance, and standard deviation.
• Probability: Measure of the likelihood that an event will occur.

Statistical Analysis

• Hypothesis Testing: Testing a claim about a population based on sample data.
• Regression Analysis: Modeling the relationship between variables.
• Confidence Intervals: Estimating population parameters with a certain level of confidence.

Mathematical Logic

• Propositional Logic: Deals with propositions (statements that are either true or false) and logical connectives (e.g., AND, OR, NOT, IF-THEN).
• Predicate Logic: Extends propositional logic to include predicates (statements about variables) and quantifiers (e.g., "for all," "there exists").
• Proof Techniques: Direct proof, indirect proof (contradiction), mathematical induction.

Set Theory

• Sets: Collections of distinct objects.
• Set Operations: Union, intersection, difference, complement.
• Venn Diagrams: Visual representation of sets and their relationships.

Functions

• Definition: A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• Types of Functions: Linear, quadratic, polynomial, exponential, logarithmic, trigonometric.
• Function Transformations: Shifting, stretching, compressing, reflecting.

Coordinate Geometry

• Cartesian Coordinate System: Using two perpendicular axes (x-axis and y-axis) to locate points in a plane.
• Distance Formula: Calculates the distance between two points in a coordinate plane.
• Slope: Measures the steepness of a line.
• Equations of Lines: Slope-intercept form, point-slope form, standard form.

Linear Algebra

• Vectors: Objects with both magnitude and direction.
• Matrices: Rectangular arrays of numbers.
• Matrix Operations: Addition, subtraction, multiplication, transposition.
• Solving Systems of Linear Equations: Using matrices and row operations.
• Eigenvalues and Eigenvectors: Important concepts in linear transformations and matrix analysis.

Discrete Mathematics

• Combinatorics: Counting techniques (permutations, combinations).
• Graph Theory: Study of graphs (networks of nodes and edges).
• Algorithms: Step-by-step procedures for solving problems.

Financial Mathematics

• Simple Interest: Interest calculated only on the principal amount.
• Compound Interest: Interest calculated on the principal and accumulated interest.
• Present Value and Future Value: Concepts used to analyze the time value of money.
• Annuities: A series of payments made at specified intervals.

Mathematical Modeling

• Process of creating a mathematical representation of a real-world situation to understand it better or make predictions.
• Involves identifying variables, formulating equations, and validating the model.

Topology

• Deals with properties of space that are preserved under continuous deformations such as stretching, twisting, crumpling, and bending, but not tearing or gluing.
• Studies concepts such as continuity, connectedness, and compactness.

Number Theory

• Branch of mathematics that deals with the properties and relationships of numbers, especially integers.
• Includes topics such as prime numbers, divisibility, and congruences.